1) Explain, using examples, the concept of “cardinality” in relation to sets.

To talk about cardinal numbers, including transfinite cardinal numbers, you need to talk about counting the numbers of elements in sets.
If a set A is finite, there is a non-negative integer, denoted |A|, which is the number of elements in A. That is one of the finite cardinal numbers.
If the set is infinite, the corresponding cardinal number is not one of the finite cardinal numbers, so it is called a transfinite (or infinite) cardinal number.

Cardinality in relation to sets, means that if sets A and B have the same number of elements, the size of the sets are the same, for example:

Let Set A = {1,3,5,7},
Set B = {apple, banana, orange, pear},
Set C = {1,2,3}

Therefore:
The cardinality of Set A, defined by |A|= 4, i.e. it contains 4 elements

The cardinality of Set B, defined by |B| = 4, i.e. it also contains 4 elements

The cardinality of Set B, defined by |C| = 3, i.e. it contains 3 elements

So we can say |A| = |B| = 4, i.e. they both have the same cardinality. In other words there is a one to one correspondence, i.e. direct mapping of each element in Set A to an element in Set B

2) How is the notion of “cardinal number” defined and how does it contrast with the notion of “ordinal number”

Cardinal numbers are numbers or symbols given to sets which display the same cardinality, that is the same symbol or number is given to those sets that have the same cardinality. The cardinality of a set is shown thus |A|or Card A.

So in the above examples, |A| = |B| = 4, i.e. both sets A and B contain 4 elements where as |C| = 3 because it contains 3 elements.

Ordinal numbers are numbers used to donate the position or the order place of a member of a ordered sequence, or precisely defined set in relation to other members of the same set.

Examples : 1st, 2nd, 3rd, 126th .

So therefore a natural number can be used for two purposes, firstly to describe the size of a set and secondly to describe the position of an element within a set.
So in the above examples:
5 is the 3rd member of Set A, banana is the 2nd element of Set B
In a finite set these two concepts coincide, but not in an infinite set. An ordinal number in set theory is one of the numbers in Cantor’s extension of the natural numbers. Finite ordinal numbers are denoted using Arabic numerals, while infinite ordinals are denoted using lower case Greek letters.

The Burali-Forti paradox dealt with ordinal numbers as opposed to the Cantor paradox, which was concerned with cardinal numbers.

3) Explain and illustrate Cantor’s paradox of Cardinal numbers.

To understand this we need to look at Cantor’s Theorem, which states that:

“The cardinal number of any set is lower than the cardinal number of the set of all its subsets” (also known as its Power set).

This can be expressed thus:

|A|  <  |P (A)| Where P(A) is the power set of A.

In other words the cardinality of the set of all sets, the Power set must be bigger than its original set - |P (A)|  >  |A|.

We can illustrate this by taking a set A, where A = {1,2,3}, therefore |A| = 3 (3 elements). The set of all its subsets is called its Power set. The Power set of A, has 2 to the power of n elements, where n is the cardinality of set A.
For example,
If set A = {1,2,3} then P (A) = {{1,2,3}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {}}
({} representing the empty set).
The Cardinality of this Power set = |P (A)| = 23 = 8 elements

Thus proving Cantor’s Theorem |A| = 3 which is less than |P (A)| = 8.

This shows that there is no largest cardinal as the cardinality of a set & its power set are not equal. So the cardinality of a set of any number of elements is less than the cardinality of its Power set.

However when this theorem is applied to the all encompassing Set of all Sets, the universal set, S, a problems arises as follows:

Firstly every subset of S is an element of S
Therefore P (S) is a subset of S
So this implies that |P (S) | <  |S|
However Cantors original theorem states that, |S| <  |P (S)|
Therefore a paradox exists.

4) Try to identify at least one proposed solution to Cantor’s paradox and illustrate how, in general terms, that solution is intended to avoid the paradox. Is this solution adequate to resolve the other paradoxes of naïve set theory investigated in the 19th and early 20th centuries.

To identify a solution to a paradox you could either show that the contradiction was only an apparent one, or that the paradox is only “proved” on invalid or unreasonable grounds.
One way the paradox can be avoided is to consider the “set” S of all sets that are not an element of itself. If one accepts the principle that all such sets can be collected into a set, then S should be a set. It is easy to see however that this leads to a contradiction (is the set S an element of itself?) . Careful choice of construction principles, so that one has the expressive power needed for usual mathematical arguments while preventing the existence of paradoxical sets. The price one has to pay for avoiding inconsistency is that some “sets” do not exist. For instance, there exists no “universal” set (the set of all sets), no set of all cardinal numbers, etc.
In the years following Cantor's discoveries, development of Set Theory proceeded with no particular concern about how exactly sets should be defined. In the early 1900s it became clear that one has to state precisely what basic assumptions are made in Set Theory; in other words, the need has arisen to axiomatize Set Theory.

The axioms for set theory now most often studied and used, although put in their final form by Skolem, are called the Zermelo-Frankel axioms (ZF). Actually, this term usually excludes the axiom of choice, which was once more controversial than it is today. When this axiom is included, the resulting system is called ZFC.
An important feature of ZFC is that every object that it deals with is a set. In particular, every element of a set is itself a set. Other familiar mathematical objects, such as numbers, must be subsequently defined in terms of sets.
The axioms of ZFC are listed below. (Strictly speaking, the axioms of ZFC are just strings of logical symbols. What follows should therefore be viewed only as an attempt to express the intended meaning of these axioms in English.)
Axiom of Extension - Two sets are the same if and only if they have the same elements.
Axiom of empty set - There is a set with no elements. We will use {} to denote this empty set.
Axiom of Pairing - If x, y are sets, then so is {x,y}, a set containing x and y as its only elements.
Axiom of union - For any set x, there is a set y such that the elements of y are precisely the elements of the elements of x.
Axiom of infinity - There exists a set x such that {} is in x and whenever y is in x, so is the union y U {y}.
Axiom of replacement - Given any set and any mapping, formally defined as a proposition P(x,y) where P(x,y) and P(x,z) implies y = z, there is a set containing precisely the images of the original set's elements.
Axiom of Power set - Every set has a power set. That is, for any set x there exists a set y, such that the elements of y are precisely the subsets of x.
Axiom of regularity - Every non-empty set x contains some element y such that x and y are disjoint sets.
Axiom of choice - Any product of nonempty sets is nonempty.
Other paradox’s which where discovered in the 19th and early 20th centuries are the Burali-Forti and the Bertrand Russell paradox’s

Ref's.

A Number for your Thoughts - Malcolm E. Lines
Schaum's Outlines - Discrete Mathematics
www.u.arizona.edu/
www.mathsworld.wolram.com
www.wkipedia.org